An Approach to Multicriteria Group Decision-Making with Unknown Weight Information Based on Pythagorean Fuzzy Uncertain Linguistic Aggregation Operators
With respect to multicriteria group decision-making (MCGDM) problems in which the experts have different priority levels, the criteria values are in the form of Pythagorean fuzzy uncertain linguistic variables (PFULVs), and the information about weights of experts and criteria is completely unknown, a novel decision-making method is developed. Firstly, the concept of PFULV is defined, and some operational laws, score function, accuracy function, and normalized Hamming distance of PFULVs are presented. Then, to aggregate information given by all experts, the Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation (PFULPWAA) operator and the Pythagorean fuzzy uncertain linguistic prioritized weighted geometric aggregation (PFULPWGA) operator are proposed. Furthermore, in order to get a comprehensive evaluation value for each alternative, the Pythagorean fuzzy uncertain linguistic Maclaurin symmetric mean aggregation (PFULMSMA) operator and the weighted PFULMSMA (WPFULMSMA) operator are proposed. Moreover, to obtain the information about the weights of criteria, the model based on grey relational analysis (GRA) method is established. Finally, a method of MCGDM with PFULVs is developed, and an application example is given to illustrate the validity and feasibility of the provided procedure.
The notion of intuitionistic fuzzy set (IFS) was presented by Atanassov , which consists of a membership degree and a nonmembership degree meeting the restriction that the sum of two degrees is equal to or less than 1. Thus, it is an effective tool to handle uncertainty and vagueness and MCGDM problems with intuitionistic fuzzy numbers (IFNs) have received more and more attention [2, 3]. Chen et al.  defined a new similarity measure between IFSs so as to solve pattern recognition problems. Based on GRA method and evidence theory, Qiu et al.  proposed a novel approach to the MCGDM problems in which both the criteria weights and criteria values take the form of IFNs. Montajabiha  developed a new version of the PROMETHE II method to solve intuitionistic fuzzy MCGDM problem. He et al.  extended power averaging operator, which can reflect the relationship between the arguments being aggregated, to IFS. He et al.  proposed some neutral aggregation operators for IFS, which reflect the interactions between IFNs and the attitude of the experts, and applied them to the MCGDM problem.
However, in some MCGDM problems, the sum of the membership degree and the nonmembership degree to which an alternative satisfies a criterion is bigger than 1, but their square sum is equal to or less than 1. Thus, the notion of intuitionistic fuzzy set of second type was presented by Atanassov [9, 10]. Later, the notion of Pythagorean fuzzy set (PFS) was presented by Yager [11, 12], which consists of a membership degree and a nonmembership degree, whose sum of squares is equal to or less than 1. Zhang  developed a novel decision method based on similarity measure to deal with selection problem of photovoltaic cells with Pythagorean fuzzy numbers (PFNs). Zhang and Xu  gave an extension of TOPSIS method to solve the MCGDM problems under Pythagorean fuzzy environment. Based on prospect theory, Ren et al.  extended TODIM method to PFS and developed an extended TODIM method. Peng and Yang  extended Choquet integral, which can consider the interactions among the criteria, to PFS, and proposed several Pythagorean fuzzy Choquet integral operators.
Under many conditions, it is difficult to handle the fuzziness and uncertainty in real MCGDM problems by numerical numbers, especially for qualitative aspects, while it is easy to express the evaluation values by means of linguistic variables (LVs) or uncertain linguistic variables (ULVs). For example, when the moral character of students, the computer performance, and so on are evaluated, they are easy to be described by the LVs, such as poor, fair, and very good. On the basis of given functions satisfying certain characteristics and distance measures, Tao et al.  defined two groups of entropies for LVs and ULVs, respectively. Liu et al.  extended Heronian mean (HM) operator, which can consider the interrelationship of the aggregated arguments, to uncertain linguistic set (ULS). Wei et al.  extended Bonferroni mean (BM) operator, which can also reflect the interrelationship of the aggregated arguments, to ULS, and applied them to the MCGDM problem with ULVs.
By combining ULVs with IFS, the concept of intuitionistic uncertain linguistic set (IULS) was introduced by Liu and Jin , which gives the information about the membership and nonmembership of an element to an ULV. Then, the research on the MCGDM problems with intuitionistic uncertain linguistic numbers (IULNs) has made many achievements [21–23]. However, the IULS cannot handle the situation; the sum of membership degree and nonmembership degree belonging to uncertain linguistic variable is bigger than 1. To deal with this situation, based on ULVs and PFS, the concept of Pythagorean fuzzy uncertain linguistic set (PFULS), which is only required to meet the restriction that the square sum of the two degrees is less than or equal to 1, is defined in this paper. To understand the PFULS better, we provide an instance: computer performance is perhaps felt to be lower than “very good” () but higher than “fair” (), the membership degree to is , and nonmembership degree is . The evaluation result can be denoted as . Due to the fact that the sum of two degrees is , the evaluation value is not available for IULS but is available for PFULS since . Clearly, the PFULS has more powerful ability than the IULS to depict the uncertainty in the real-world MCGDM problems. It should be noted that when the upper and lower limits of the uncertain linguistic part of PFULS are identical, PFULS reduces to the Pythagorean fuzzy linguistic set (PFLS) introduced by Peng and Yang , which indicates that the former is an extension of the latter. Compared with PFLS, PFULS is defined by utilizing ULVs, whose membership degree and nonmembership degree are no longer with respect to a LV, but to an ULV, which makes the experts express uncertain information more easily and precisely.
Information aggregation is a pervasive activity in our daily life; many operators have been provided on this issue. Among them, the prioritized averaging (PA) operator and the Maclaurin symmetric mean (MSM) are two of the most common operators for aggregating information. The PA operator was initially presented by Yager , which can capture the prioritization phenomenon of the aggregated arguments. Then, it was extended to hesitant fuzzy set (HFS) , triangular fuzzy set (TFS) , IFS , trapezoidal intuitionistic fuzzy set (TIFS) , linguistic set (LS) , 2-tuple linguistic set (2TLS) , multigranular uncertain linguistic set (MULS) , and so on. The MSM was initially given by Maclaurin , which can consider the interdependent characteristics among the multi-input arguments. The MSM is different from Choquet integral or power average operator. The MSM pays attention to the input arguments while the Choquet integral or power average operator pays attention to the weights information. The MSM is also different form BM or HM. The MSM operator considers the interdependent characteristics among the multi-input arguments and should take one parameter from finite integer set, while the BM or HM captures the interrelationship between two input arguments and should take two parameters both from infinite set. Due to the advantages of the MSM, it was extended to HFS , IFS , 2TLS , ULS , intuitionistic linguistic set (ILS) , IULS , and so on. However, both PA and MSM operators fail to aggregate Pythagorean fuzzy uncertain linguistic information. Therefore, we shall propose the PFULPWAA, PFULPWGA, PFULMSMA, and WPFULMSMA operators. The significant features of these operators are that not only can they handle PFULVs, but also the PFULPWAA and PFULPWGA operators can capture prioritization among the criteria or experts, and the PFULMSMA and WPFULMSMA operators can consider the interrelationship among the multi-input arguments.
The aim of this paper is to develop a novel method based on proposed Pythagorean fuzzy uncertain linguistic aggregation operators to solve the MCGDM problems in which the experts have different priority levels, the criteria values are in the form of PFULVs, and the information about weights of experts and criteria is completely unknown. To do so, the remainder of this paper is constructed as follows: Section 2 reviews some concepts of PFS, PA operator, and MSM. Section 3 defines the concept of PFULS and presents some operational laws, score function, accuracy function, and normalized Hamming distance of PFULVs. Section 4 proposes the PFULPWAA and PFULPWGA operators and investigates their corresponding properties. Section 5 proposes the PFULMSMA and WPFULMSMA operators. Section 6 presents an approach to Pythagorean fuzzy linguistic MCGDM based on GRA model and the proposed new operators. Section 7 provides an example to demonstrate the decision-making application. Section 8 gives the concluding remarks.
2.1. The Pythagorean Fuzzy Set
Definition 1 (see ). Suppose is a universe of discourse. A Pythagorean fuzzy set (PFS) in is an expression given by where the functions and are the degree of membership and the degree of nonmembership of the element to the set , respectively, meeting the condition that . The function is called the degree of indeterminacy of to the set . For computational convenience, is called a Pythagorean fuzzy number (PFN) represented by .
With respect to the operational rules and characteristics of PFNs, please refer to [11, 14].
2.2. The Prioritized Averaging Operator
In many real and practical MCGDM problems, the criteria or the experts usually have different priority levels. For instance, regarding decision-making in a company, general manager usually has a higher priority than vice manager. To deal with this issue, the prioritized averaging (PA) operator was proposed by Yager , which is shown as follows.
Definition 2 (see ). Suppose that the criteria are prioritized, where (“” denotes “be superior to”). The value is the criterion value of any alternative under criterion and meets . Ifwhere , . Then, PA is called the prioritized averaging operator.
2.3. The Maclaurin Symmetric Mean
The MSM can consider the interdependent characteristics among the multi-input arguments, which is shown as follows.
Definition 3 (see ). Suppose is a collection of nonnegative real numbers, and . Ifwhere is the binomial coefficient, and traverses all the -tuple combinations of , then, is called the Maclaurin symmetric mean.
3. Pythagorean Fuzzy Uncertain Linguistic Variable
3.1. The Pythagorean Fuzzy Uncertain Linguistic Set
Motivated by PFLS  that is based on LVs and PFS, we define the notion of PFULS based on ULVs and PFS and present operational laws, score function, accuracy function, and normalized Hamming distance of PFULVs.
Definition 4. Suppose is a universe of discourse. A PFULS in is an expression given by where is an ULV defined by Xu  and is a PFS expressed in Definition 1 which denotes the degrees of the element to the ULV . The function is called the degree of indeterminacy of to the ULV For computational convenience, is called a Pythagorean fuzzy linguistic variable (PFULV) represented by .
Suppose that and are any two PFULVs and ; then based on operational laws of ULVs and PFNs, the operational laws of and are given as follows:Clearly, the above operational results are still PFULVs.
Theorem 5. Suppose that and are any two PFULVs; then we can get the following calculation rules:
Proof. (1) According to formulas (5) and (6), formulas (9) and (10) are right.
(2) For formula (11),and the proof of formula (11) is finished.
(3) Similar to proof of formula (11), it can be easily proved that formulas (12), (13), and (14) are right.
3.2. Comparison of Two Pythagorean Fuzzy Uncertain Linguistic Variables
Definition 6. Suppose that is a PFULV; then the score function of is shown as follows:It should be mentioned that the score function is between and . In order to facilitate the following study, we provide another score function whose range is between and .
Definition 7. Suppose that is a PFULV; then the accuracy function of is shown as follows:
Definition 8. Suppose that and are any two PFULVs; then their comparison rules are shown as follows:(1)If , then .(2)If and , then .(3)If and , then
3.3. The Normalized Hamming Distance between Two PFULVs
Definition 9. Let , , and be any three PFULVs. If satisfies the three restrictions(1),(2),(3),then is called the distance between and .
Definition 10. Suppose that and are any two PFULVs; then the normalized Hamming distance between and is defined as follows:
Proof. Clearly, the normalized Hamming distance defined in (19) satisfies restrictions (1) and (2) in Definition 9. In what follows, we shall prove that the normalized Hamming distance defined in (19) can also satisfy restriction (3) in Definition 9.and therefore
4. Pythagorean Fuzzy Uncertain Linguistic Prioritized Aggregation Operators
In this section, we shall extend the PA operator to PFULS and propose the PPFULPWAA and PFULPWGA operators.
4.1. The PFULPWAA Operator
Definition 11. Suppose is the set of all PFULVs, is a collection of PFULVs, and ; ifwhere , and is the score value of a PFULV , then PFULPWAA is called the Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation operator.
Theorem 12. Suppose that is a collection of PFULVs; then the aggregating value by PFULPWAA operator is still a PFULV, and where , , and is the score value of the
Proof. Formula (23) can be proved by mathematical introduction on as follows.
(i) For , according to the operational laws of PFULVs, we obtainSo, formula (23) is right for .
(ii) Suppose that formula (23) is also right for ; that is,(iii) For , we obtainSo, formula (23) is right for . According to (i), (ii), and (iii), formula (23) is right for all .
In addition, sinceformula (23) is still a PFULV, and the proof of Theorem 12 is finished.
In what follows, some properties of the PFULPWAA operator shall be explored.
Theorem 13. Suppose that is a collection of PFULVs; then the PFULPWAA operator has the following properties:(1)(Idempotency). Suppose all are equal; that is, , and then(2)(Boundness). Suppose , and ; then
Proof. (1) According to Definition 11 and formula (13) in Theorem 5, we obtain(2) Since , , , and , then we obtainFurthermore, according to Definition 6, we obtainThus, we should analyze the following two situations.
(1) If , according to Definition 8 and the idempotency of the PFULPWAA operator, we obtain (2) If , according to formulas (31), we obtain and according to Definition 7, we obtainthen according to Definition 8 and the idempotency of the PFULPWAA operator, we obtain Therefore, we can obtain .
Similarly, we have .
Therefore, we obtain .
4.2. The PFULPWGA Operator
Based on PFULPWAA operator and geometric mean, the PFULPWGA operator is defined as follows.
Definition 14. Suppose is the set of all PFULVs, is a collection of PFULVs, and , and ifwhere , and is the score value of a PFULV , then PFULPWGA is called the Pythagorean fuzzy uncertain linguistic prioritized weighted geometric aggregation operator.
Theorem 15. Suppose that is a collection of PFULVs; then the aggregating value by PFULPWGA operator is still a PFULV and where , and is the score value of the
5. Pythagorean Fuzzy Uncertain Linguistic Maclaurin Symmetric Mean Aggregation Operators
In this section, we will extend the MSM to PFULS and propose the PFULMSMA and WPFULMSMA operators.
5.1. The PFULMSMA Operator
Definition 16. Suppose is the set of all PFULVs, is a collection of PFULVs, , and , and ifwhere is the binomial coefficient and traverses all the -tuple combinations of , then PFULMSMA is called the Pythagorean fuzzy uncertain linguistic Maclaurin symmetric mean aggregation operator.
Theorem 17. Suppose is a collection of PFULVs, and ; then the aggregating value by PFULMSMA operator is still a PFULV, and
In what follows, some properties of the PFULMSMA operator will be explored.
Theorem 18. Suppose that and are two collections of PFULVs, and , and then the PFULMSMA operator has the following properties:(1)(Idempotency). Suppose all are equal; that is, , and then(2)(Commutativity). Suppose is any permutation of ; then(3)(Monotonicity). Suppose ; that is, , , , for all , and then(4)(Boundness). Suppose and ; then